Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 303 / 1 / b / i / Solution

Let $z=2d$ be the coordination number of the hypercubic lattice. The mean interaction energy per site is $(Jz/2)m_e{m}_o=Jd,m_e{m}_o$. DefineThe two sublattices each contain half the sites, so the mean-field free energy per site isThe entropy terms are the binary-spin mixing entropies and the positive $J$ favors opposite sublattice magnetizations.